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Spheroidal WaveFunctions in

Electromagnetic Theory

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Spheroidal WaveFunctions in

Electromagnetic Theory

Le-Wei Li

Xiao-Kang Kang

Mook-Seng Leong

A Wiley-Interscience Publication

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Li, Le-Wei.Spheroidal wave functions in electromagnetic theory / Le-Wei Li, Xiao-Kang Kang,

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ISBN 0-471-03170-4 (cloth : alk. paper)1. Electromagnetic theory. 2. Spheroidal functions. I. Kang, Xiao-Kang. II. Leong,

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QC670.L492001530.14'1—dc21 2001045399

Printed in the United States of America

1 0 9 8 7 6 5 4 3 2 1

Preface

Spheroidal coordinates and spheroidal wave functions have found many im-portant and practical applications in electromagnetic theory, such as antennaanalysis and design, MIC design, EMC and EMI, and radar cross sectionscalculations. There are several hot topics in computational electromagneticsrelated to spheroidal structures. For instance, rockets, aircraft noses, andguided missiles are generally considered to have spheroidal shapes. Humanheads can be modeled approximately as prolate spheroids in the calculation ofelectromagnetic interaction between a head and a cellular phone. Raindropscan be modeled as oblate spheroids in computation of the rainfall attenuationof microwave signals in line-of-sight and satellite telecommunication systems.The theory of spheroidal wave functions was developed about half a centuryago. However, applications of these functions were not widely made in var-ious research areas involving electromagnetics. Apparently, there are manyfar fewer reports in the literature on full-wave analysis of electromagneticscattering and radiation in spheroidal coordinates than those in other coor-dinate systems, for instance, cylindrical and spherical coordinates. One ofthe difficulties is the non-existence of orthogonality among spheroidal wavefunctions. The prolate and oblate spheroidal coordinates are two systems inwhich the scalar wave equations are separable but the vector wave functionsare not separable. This therefore causes the difficulty of obtaining rigoroussolutions of those vector boundary value problems. Another challenge is thevery complicated calculation of the spheroidal harmonics, especially for those

v

vi PREFACE

spheroids with large focal distances and complex dielectric properties in thehigh-frequency regions.

This book presents in detail the theory of spheroidal wave functions, ap-plies it widely to the analysis of electromagnetic fields in various spheroidalstructures, and provides a comprehensive programming code for those compu-tations. In addition to the theory to be detailed and applications to be coveredwidely, several Mathematica package source codes are developed for the cal-culation of spheroidal wave functions and for their practical applications ofthose functions in electromagnetic theory.

The following topics are covered in the this book: (1) theory of the eigen-value problem, spheroidal harmonics, and spheroidal vector wave functions;(2) electromagnetic scattering by a conducting spheroid and a dielectric-coated conducting spheroid; (3) electromagnetic radiation from a conductingspheroidal dipole antenna, a dielectric-coated conducting spheroidal dipoleantenna, and a dielectric-coated conducting spheroidal dipole antenna cov-ered with a dielectric spheroidal radome shell; (4) dyadic Green's functionsin single- and multi-layered spheroidal structures; and (5) the specific absorp-tion rate (SAR) of power due to dipole radiation into a multilayered spheroidalhuman head. A Mathematica software package for computing spheroidal func-tions is developed on the basis for purposes of comparative study and pro-vided partially on the Web site (http://www.wolfram.com/mathsource) andcompletely on the Web site (http://www.ece.nus.edu.sg/lwli).

Proposed readers of this book are those scientists, engineers, and graduatestudents in electromagnetics, applied physics, or applied mathematics whoseresearch work involves spheroidal coordinates or models. It is hoped that theapproach of this book will help readers to understand theory and techniquesused for prolate and spheroidal structures and other nonorthogonal systems.It is, then, necessary that readers take full account of the theory and the soft-ware routine packages in this book, if they wish to find further solutions ofcomplicated structures by using the attached source codes. Those scientistsand engineers interested in some of the applications, such as SAR Distribu-tions in a Spheroidal Head Model and Rain Attenuation of Oblate SpheroidalRaindrop, can also use the numerical values there as further references.

LE-WEI LIXIAO-KANG KANG

MOOK-SENG LEONG

National University of Singapore

Acknowledgments

This book is a part of research results of a Joint Research Project sponsoredby National University of Singapore and DSO National Laboratories, Singa-pore. In this connection, the authors wish to acknowledge DSO National Labsfor the financial support and especially our colleague, Mr. Yeow-Beng Gan atTemasak Laboratory at the National University of Singapore, for his strongencouragement and help. Thanks also go to our other colleagues, ProfessorPang-Shyan Kooi and Professor Tat-Soon Yeo of the Department of Electri-cal and Computer Engineering at the National University of Singapore, whoshowed their encouragement and support in one way or another. The firstauthor also wishes to express acknowledgment to Dr. Paul C. Abbott ofthe Department of Physics at University of Western Australia for his usefuldiscussions and suggestions during the preparation of the book.

The authors would also like to acknowledge our graduate students who pro-vided assistance in the software development of Mathematica packages, someof the numerical results in Chapters 4, 5 and 6 of this book, and proofreading,including Mr. Kian-Yong Tan, Mr. Chia-Heng Quek, Mr. Phin-Juay Loh,and Mr. Zhong-Cheng Li.

We would like to extend our gratitude to the Wiley-Interscience Series Edi-tor: Professor Kai Chang at Texas A&M University for his encouragement, tothe technical reviewers: Professor Qing-Huo Liu at Duke University and Pro-fessor Jian-Ming Jin at University of Illinois at Urbana-Champaign for theirinvaluable suggestions and strong recommendations, to the Executive Editorof Wiley-Interscience and the Managing Editors of Wiley STM Production:

vii

viii ACKNOWLEDGMENTS

George J. Telecki, Andrew Prince and Angioline Loredo for their interest inthe publication and careful copy-editing of this book, and to the EditorialAssistant of Wiley-Interscience: Sara A. Paracka for her professional help.

Finally, taking this opportunity, the authors wish to acknowledge our belovedwives and children for their understanding, encouragement, and support inone way and another.

LW Li, XK Kang, and MS Leong

Contents

Preface v

Acknowledgments vii

1 Introduction 11.1 Overview 11.2 EM Scattering by Spheroids 41.3 Spheroidal Antenna 51.4 EM Radiation in Dielectric Spheroids 71.5 Oblate Spheroidal Models 81.6 Spheroidal Cavity System 91.7 Spheroidal Harmonics and Mathematica Software 10

2 Spheroidal Coordinates and Wave Functions 132.1 Spheroidal Coordinate Systems 132.2 Spheroidal Scalar Wave Functions 172.3 Spheroidal Angular Harmonics 18

2.3.1 Series Representation in Terms ofAssociated Legendre Functions 18

2.3.2 Power Series Representation 202.4 Eigenvalues mn and Expansion Coefficients dmn

r 22

ix

x CONTENTS

2.4.1 Case I: |c|2 < 1000 232.4.2 Case II: |c|2 > 1000 26

2.5 Spheroidal Radial Harmonics 272.5.1 Series Representation in Terms of

Spherical Bessel Functions 272.5.2 Proportional Relations of Angular and

Radial Functions 292.5.3 Power and Legendre Functional Series

Representations 302.6 Derivatives of Spheroidal Functions 35

2.6.1 Derivatives of Angular Functions 352.6.2 Derivatives of Radial Functions 35

2.7 Numerical Calculations and Discussion 362.7.1 Mathematica Source Codes 362.7.2 Geometrical Features of Spheroidal

Functions 372.7.3 Tabulated Numerical Data: New Results

and Comparison 372.8 Spheroidal Vector Wave Functions 44

3 Dyadic Green's Functions in Spheroidal Systems 613.1 Dyadic Green's Functions 613.2 Fundamental Formulation 633.3 Unbounded Dyadic Green's Functions 66

3.3.1 Method of Separation of Variables 663.3.2 Unbounded Scalar Green's Function 673.3.3 Appropriate Spheroidal Vector Wave

Functions for Construction of DGFs 683.3.4 Unbounded Green's Dyadics 69

3.4 Scattering Green's Dyadics 703.4.1 Scattering Green's Dyadics in the Inner

Region (f = l) 713.4-2 Scattering Green's Dyadics in the

Intermediate Regions (2 < f < N - l) 713.4-3 Scattering Green's Dyadics in the Outer

Region (f = N) 723.5 Determination of Scattering Coefficients 73

3.5.1 Nonorthogonality and FunctionalExpansion 73

3.5.2 Matrix Equation Systems 76

CONTENTS xi

3.6 Convergence of the Solution 86

4 EM Scattering by a Conducting Spheroid 894-1 Geometry of the Problem 894-2 Incident and Scattered Fields 894-3 Transformation of Incident Fields to Scattered

Fields 924-3.1 Imposing the Boundary Conditions 924.3.2 TE Polarization for Oblique Incidence 934.3.3 TM Polarization for Oblique Incidence 994.3.4 Fields at Axial Incidence 1014.3.5 TE Fields with Incidence Angle 90° 102

4-4 Far-Field Expressions 1034.5 Numerical Computation and Mathematica Source

Codes 1064.6 Results and Discussion 108

5 EM Scattering by a Coated Dielectric Spheroid 1155.1 Geometry of the Problem 1155.2 Incident, Transmitted and Scattered Fields 1175.3 Relationship between Incident and Scattered

Fields 1195.3.1 Boundary Conditions 1195.3.2 TE Polarization for Nonaxial Incidence 1195.3.3 TM Polarization for Nonaxial Incidence 1285.3.4 Fields at Axial Incidence 130

5.4 Numerical Computation and Mathematica SourceCode 130

5.5 Results and Discussion 132

6 Spheroidal Antennas 1456.1 Introduction 1456.2 Prolate Spheroidal Antenna 146

6.2.1 Antenna Geometry 1466.2.2 Maxwell's Equations for the Spheroidal

Antenna 1466.2.3 Auxiliary Scalar Wave Function 1486.2.4 Imposing the Boundary Conditions 1496.2.5 Far-Field Expressions 150

xii CONTENTS

6.2.6 Numerical Computations and MathematicaCode 150

6.2.7 Results and Discussion 1516.3 Dielectric-coated Prolate Spheroidal Antenna 152

6.3.1 Coated Dielectric Antenna Geometry 1526.3.2 Obtaining the Auxiliary Wave Functions 1586.3.3 Imposing the Boundary Conditions 1616.3.4 Numerical Computations 1626.3.5 Mathematica Code 1636.3.6 Results and Discussion 165

6.4 Prolate Spheroidal Antenna enclosed in aConfocal Radome 1686.4.1 Geometry of the Antenna with Radome 1686.4-2 Obtaining the Auxiliary Wave Functions 1746.4.3 Imposing the Boundary Conditions 1746.4-4 Numerical Computations 1766.4.5 Mathematica Code 1776.4.6 Results and Discussion 179

7 SAR Distributions in a Spheroidal Head Model 1917.1 Introduction 1917.2 Multilayered Prolate Spheroidal Head Model 1927.3 Formulation of the Problem 194

7.3.1 Expansions of EM Fields Using SpheroidalWave Functions 194

7.3.2 EM Boundary Conditions for MultispheroidalInterfaces 195

7.3.3 Specific Absorption Rate 1957.4 Numerical Computation 1967.5 Results and Discussion 1977.6 Effects on Wire Antennas Due to the Presence

of the Multilayered Spheroid 2097.7 Numerical Results and Discussion 218

8 Analysis of Rainfall Attenuation Using Oblate Raindrops2278.1 Introduction 227

8.1.1 Rainfall Attenuation 2278.1.2 Raindrop Models in Different Sizes 2288.1.3 Oblate Spheroidal Raindrops 229

CONTENTS xiii

8.2 Problem Formulation 2308.2.1 Geometry of the Problem 2308.2.2 Definition of the EM Field 2308.2.3 Boundary Conditions and Solution of

Unknowns 2348.2.4 Total Cross Section 237

8.3 Size Parameters of Raindrops 2388.3.1 Radius-Independent Oblate Spheroid

Raindrop 2388.3.2 Radius-Dependent Oblate Spheroid

Raindrop 2388.4 Numerical Calculation and Results 239

9 EM Eigenfrequencies in a Spheroidal Cavity 2459.1 Introduction 2459.2 Theory and Formulation 246

9.2.1 Background Theory 2469.2.2 Derivation 247

9.3 Numerical Results for TE Modes 2499.3.1 Numerical Calculation 2499.3.2 Results and Comparison 250

9.4 Numerical Results for TM Modes 2529.4.1 Numerical Calculation 2529.4.2 Results and Comparison 252

9.5 Discussion 254

Appendix A Expressions of Spheroidal Vector Wave Functions255

Appendix B Intermediates It,l™n(c) in Closed Form 263

B. 1 The Case where m > 1 264B.2 The Case where m = 0 269

Appendix C Uq(i) , t and vq(i),t Used in the Matrix EquationSystem 273

References 217

Index 292

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Spheroidal WaveFunctions in

Electromagnetic Theory

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1Introduction

1.1 OVERVIEW

Spheroidal wave functions are special functions in mathematical physics whichhave found many important and practical applications in science and engineer-ing where the prolate or the oblate spheroidal coordinate system is used. Inthe evaluation of electromagnetic (EM) fields in spheroidal structures, spher-oidal wave functions are frequently encountered, especially when boundaryvalue problems in spheroidal structures are solved using full-wave analysis.By applying the separation of variables to the Maxwell's equations satisfiedby either an electric or magnetic field, the spheroidal harmonics of electro-magnetic waves, corresponding to their spheroidal coordinate system, can beobtained.

With prolate or oblate spheroidal coordinates, the separation of scalar vari-ables results in three independent functions: (1) the radial spheroidal functionRmn(c,£) or R$n(—ic, i£) , (2) the angular spheroidal function Smn(c, T?) orSmn(~ic,T)), and (3) the sine and cosine functions. This separability is anal-ogous to that of solving the Laplace equation in spherical coordinates. Herethe last pair of trigonometrical functions (sine and cosine) is well known, butthe first two are not so easily computed. In general, spheroidal radial and an-gular functions in spheroidal coordinates are, respectively, the generalizationof Legendre functions and spherical Bessel functions in the spherical polarcoordinates [6]. Computation of the spheroidal radial or angular functionsrequires the eigenvalue computation and the forward and backward recur-

1

2 INTRODUCTION

sion formulations. Theoretically, the formulation of these harmonics was welldocumented by J. A. Stratton et al. in 1956 [7] and C. Flammer in 1957 [1].

So far, there are only six coordinate systems in which the scalar Helmholtzequations are separable and solenoidal solutions of the vector Helmholtz equa-tions which are transverse to coordinate surfaces are obtainable: the rectan-gular, the circular-cylinder, the elliptic-cylinder, the parabolic-cylinder, thespherical, and the conical coordinate systems [8,9]. The prolate and oblatespheroidal coordinate systems are two systems in which scalar wave equationsare separable but the vector wave functions are not separable because of thenonorthogonality of spheroidal radial functions. This causes a difficulty inobtaining rigorous solutions to those vector boundary value problems.

In computing EM or physical quantities, the nonorthogonality is involvedin the method of eigenfunctional expansions, in which the linear equationsin the form of nonorthogonal summation are converted to a matrix equationsystem. The dimension of the matrix system is normally infinite and the con-vergence depends mostly on the magnitudes of the interfocal distance and theapplied frequency range. The convergence also depends on the representationof different kinds of spheroidal wave functions. Several kinds of spheroidalvector wave functions have been introduced by researchers [10-12]. However,most of them show fast convergence only for plane-wave scattering or far-fieldapproximations in free space. When the dielectric properties of propagationmedia are lossy, the convergences of some kinds of spheroidal vector wavefunctions become very slow. Furthermore, some kinds of spheroidal vectorwave functions converge much more slowly in the high-frequency region, al-though it is possible to use these vector wave functions in the low-frequencyregion.

When used to solve practical problems, the most appropriate spheroidalvector wave functions are constructed by using the spheroidal scalar eigen-functions as the generating function and the orthogonal coordinate vectors(such as the unit vector in the rectangular coordinates) as the piloting vector[13]. Construction of the spheroidal vector wave functions is similar to thatdescribed by Tai [14] for orthogonal coordinates (such as the rectangular orspherical wave functions), except that the spheroidal vector wave functionscannot simply be separated to those of transverse electric (TE) or transversemagnetic (TM) modes. It is found in practical applications that this kind ofspheroidal vector wave function is very convenient in calculations involving in-tegral equation expressions, especially for cases where the electric or magneticsources have arbitrary shapes.

One of the most convenient analytical methods of EM problems is theintegral representation of wave equations, in which the Green's function is de-sirable. To obtain electromagnetic radiation due to an arbitrary current dis-tribution located in an inhomogeneous medium, the dyadic Green's functiontechnique is usually adopted. If the source is of unknown current distribution,the method of moments, which expands the current distribution into a seriesof basis functions with unknown coefficients, can be employed. In this case,

OVERVIEW 3

the dyadic Green's function is considered as a kernel of the integral. Therelated unknown coefficients of the basis functions can be obtained in matrixform by enforcing the boundary conditions to be satisfied. The constructionof Green's dyadics in spheroidal structures using the appropriate spheroidalscalar eigenfunctions is presented by Li et al. [15].

Theoretically, spheroidal structures can be reduced to spherical structureswhen the interfocal distance d becomes zero or the radial coordinate becomesinfinity (while 77 = constant), as shown in Figs. 1.06 and 1.07 in [8]. For asmall-value parameter d, the spheroid can be well approximated by a sphere.In comparing spherical and spheroidal functions, deviations from the spheri-cal functions increases as d increases. It is clear that for a large d, sphericalapproximation is not valid. Some analytical results of EM fields in spher-oidal structures have been obtained using spherical vector harmonics [16,17];however the convergences are generally slow and the results are less accuratewhen the interfocal distances of spheroids become larger. Even in some casesof spheroids with small d values, numerical results from the spherical approx-imation are also not accurate for the near-field calculation or in the very highfrequency region. This is caused by the fact that more higher-order harmon-ics are needed in the summation for near-field or high-frequency expressions.The convergence condition of the spherical wave functions approximation insuch cases becomes worse than that of spheroidal eigenfunction expansions.

Besides the lack of orthogonality of spheroidal vector wave functions, thevery complicated calculation of the spheroidal angular and radial harmonics isanother difficulty in obtaining the analytical solutions in spheroidal structures[1,5]. This is part of the reason that there have been many fewer reports aboutthe applications of spheroidal wave functions in computational electromagnet-ics than those related to other coordinate systems, although the spheroidalwave functions have been known for more than 100 years. It is also found thatsome of the previous computed results of spheroidal eigenvalues and spher-oidal harmonics are not accurate [5], although they have been used elsewherefor many years [1,18].

With the development of computer facilities and the appropriate methodand software routines for calculation [2,19], the numerical values tabulatedfor spheroidal angular and radial functions can be obtained easily and accu-rately. The analysis of spheroidal systems can be carried out in the sameway as in the spherical system, by using general integral equation expressionswith the formulated spheroidal Green's functions and related spheroidal wavefunctions, while the functional expansions should be employed in spheroidaleigenfunction expansions. In practical programming, the spheroidal angularand radial functions can be treated similarly to Legendre and Bessel functionsin spherical coordinates and obtained from available numerical tables or sub-routine calling of specific packages. Prepared numerical tables are normallypreferred to the repetitive calculation of one spheroidal structure. Therefore,many electromagnetic problems related to spheroidal structures can finally be

4 INTRODUCTION

solved directly in spheroidal wave function expansions by full-wave analysismethods.

There have been an increasing number of applications of spheroidal wavefunctions in computational electromagnetics. In antenna designs, rockets,satellites, and guided missiles can be considered to have part of a spher-oidal shape. The related EM wave scattering problems have been studiedfor many years. In some applications, microstrip antennas are mounted oncurved spheroidal surfaces. With the development of mobile communications,spheroidal dipoles and monopoles have been investigated intensively. Also,electromagnetic interaction between the human head and a cellular antennaand the health effects of mobile phone have been hot topics in recent years.In these EM problems, the human head can be approximated as a dielectricprolate spheroid. In the computation of the rainfall attenuation of microwavesignals, a raindrop can be modeled as an oblate spheroid, especially for rain-drops of large sizes. In this book, we provide a generalized calculation anddiscussion regarding the application of spheroidal wave functions in these EMproblems and attach our own developed software packages.

1.2 EM SCATTERING BY SPHEROIDS

EM waves scattered by a single spheroid or a system of n spheroids have beenwell investigated, and many analytic solutions have been obtained to date.Jen analyzed the field and current distribution of a prolate spheroidal dipoleantenna embedded in a larger dielectric confocal prolate spheroid with finiteconductivity [20]. He also gave a theoretically analyzed solution of the radia-tion due to a thin linear monopole of arbitrary length erected at the tip andalong the axis of a metallic prolate spheroid of any length and eccentricity[21]. Moffatt studied the echo area of a perfectly conducting prolate spheroid[22,23]. The approximate solution to the EM-wave backscattering from a pro-late spheroid was obtained through modification of the impulse response ortime-dependent backscattering waveform from a perfectly conducting spheri-cal scatterer.

Asano and Yamamato provided a solution of EM scattering by a homo-geneous prolate (or oblate) spheroidal particle with an arbitrary size andrefractive index at any angle of incidence [24]. The method they used wasthat of separating the vector wave equations in the spheroidal coordinatesand expanding them in terms of the spheroidal wave functions. The methodof determining the unknown coefficients for the expansion was similar to thatused in this book. However, Asano and Yamamato's report only showed thenumerical results for c no larger than 10 (where c is the product of the wavepropagation constant and semi focal distance). Asano employed the samemethod to determine that the prolate spheroids at parallel incidence havesteep and high resonance maxima in scattering efficiency factors, and broadand low forward-scattering peaks in the intensity functions. On the other

SPHEROIDAL ANTENNA 5

hand, oblate spheroids at parallel incidence have broad and low resonancemaxima and sharp and high forward-scattering peaks [25]. It was shown thatat oblique incidence, the scattering properties of a long, slender prolate spher-oid resemble those of an infinitely long circular cylinder [25].

Sinha and MacPhie also used spheroidal wave functions to characterizescattering of plane waves with arbitrary polarization and angle of incidenceby conducting prolate spheroids [10,11]. In their formulation, the columnmatrix of the series coefficients of the scattered field is obtained from thecolumn matrix of the series coefficients of the incident field by means of amatrix transformation. The matrix depends only on the scatterer, hence thescattered field for a new direction of incidence is obtained without repeatedlysolving a new set of linear equations. This method was employed by someother researchers and had many applications in physics and engineering.

Dalmas and Deleuil analyzed the multiple scattering of EM wave by twoinfinitely conducting prolate spheroids, using the spheroidal vector wave func-tions constructed along the radius vector [26]. The formulation which theyused requires the establishment of a translational addition theorem for thewave functions of any translation [27]. Merchant et al. used Waterman'stransition-matrix (T-matrix) method to obtain the complex pole patterns ofthe scattering amplitude for conducting spheroids [28].

Cooray and Ciric presented their solutions of EM wave scattering by asystem of two perfectly conducting or dielectric spheroids of arbitrary orien-tation [12,29]. The method they used was similar to that of Sinha and Mac-Phie [10,11], In order to impose the boundary conditions, the field scatteringby one spheroid is expressed in terms of the spheroidal coordinates attachedto the other spheroid, by using the rotational-translational addition theoremfor spheroidal vector wave functions [30]. The rotational-translational addi-tion theorems were also verified recently by Nag and Sinha solving the EMscattering by a system of two lossy dielectric prolate spheroids [31].

Using the same method as in [30], Cooray and Ciric solved the problems ofEM scattering by a system of n dielectric spheroids of arbitrary orientation [32]and EM wave scattering by a coated dielectric spheroid [33]. Sebak and Sinhaprovided the solution of a plane EM wave scattering by a conducting prolatespheroid coated with a confocal homogeneous layer [34]. In their research,the incident and scattered fields were expanded in terms of spheroidal vectorwave functions and the expansion coefficients for the scattered wave weredetermined directly by the application of boundary conditions.

1.3 SPHEROIDAL ANTENNA

Interests in spheroidal wave functions and spheroidal antennas have come along way. Such an interest is understandable, as spheroidal antennas can beused to model a variety of antenna shapes, from wire/cylindrical antennas viaspherical antennas to disk antennas (using oblate spheroid).

6 INTRODUCTION

One of the earliest analyses for the spheroidal antenna problem was doneby Schelkunoff [35]. In his work, he analyzed the prolate spheroidal antennawhere the voltage is applied symmetrically between two halves of the spher-oid by means of a biconical transmission line. In this case, the far field isindependent of the coordinate. The input admittances for such antennas areobtained for various major to minor axial ratios and the first few spheroidalmodes of propagation.

Chu and Stratton investigated the boundary value problem of the forcedEM oscillations of a conducting prolate spheroid which was fed by a gap ofinfinitesimal width across the central section of the spheroid, while only fewmodes were considered in the investigation [36], Flammer also analyzed theradiation characteristics of a prolate spheroidal monopole antenna with a finitegap using a variational approach [37].

Weeks [38] analyzed the problem of the prolate spheroidal antenna from aslightly different angle. Instead of having the excitation gap in the centre ofthe antenna, he allowed the gap to be located arbitrarily along the antennawithout disturbing the symmetry in the direction. In addition, he also solvedthe problem of a dielectric-coated spheroidal antenna as a boundary valueproblem. His results were presented in the form of radiation patterns.

Some authors also used the ultraspherical distribution of cylindrical dipoleantennas as an extension to study the spheroidal dipoles [39]. Do-Nhat andMacPhie studied the input admittance of thin prolate spheroidal dipole anten-nas with finite gap width [39]. In their study, different kinds of gap fields withvarious gap widths produce the same admittance value for very thin spheroidsbut different values of the admittance for thicker spheroidal antennas. It hasbeen found that for prolate spheroidal antennas, the first resonance occurswhen their lengths are slightly greater than half wavelength, in contrast tocircular cylindrical antennas which resonate when their lengths are slightlyless than half a wavelength.

Jen and Hu [40] presented the radiated fields and input admittance of ametallic prolate spheroid excited by a circumferential slot. It is found thatthe far-zone radiation patterns depend more on the current distribution thanon the position of the exciting slot. In their research, they treated spheroidalfunctions of large c values, using the asymptotical calculation of spheroidalscalar functions by means of the Wentzel-Kramer-Brillouin and Langer trans-formations.

Vinogradova investigated the radiation characteristics of a surface non-closed spheroidal antenna using the spheroidal scalar wave functions [41].The radiation pattern of the nonclosed spheroidal antenna varies with thefrequency in a narrow band. Kotulski [42] computed the input admittance ofthe prolate spheroidal antenna by using an equivalent circuit for the first twomodels of the antenna.

Zhang and Sebak [43] studied the radiation characteristics of an asymmet-rical slot antenna on a conducting prolate spheroid. In their work, the vectorprolate spheroidal wave functions were used to expand the radiated fields.

EM RADIATION IN DIELECTRIC SPHEROIDS 7

Using a system of equations derived from the boundary conditions, they wereable to obtain the unknown expansion coefficients. The effects of the slotlength and shape of the spheroid on the radiated field were also presented.

All of the studies above deal with problems of a single antenna. The studyof coupled spheroidal antennas has also received interest from researchers.Sinha and MacPhie [44] applied the translational addition theorem for spher-oidal wave functions to investigate the problem of a two-element parallel pro-late spheroidal antenna system. The same method was used by Ciric andCooray [45] to calculate the admittance characteristics and far-field patternsfor coupled spheroidal dipole antennas in arbitrary orientation. The calcula-tion of the self and mutual admittances can be applied to reduce the couplingbetween the two spheroidal antennas for various orientations.

With the development of mobile communication systems and iridium satel-lite communication systems, the microstrip antenna is widely used, as it isadaptive in size, thin and lightweight, and has larger gain than that of wireantennas. In some practical applications or to widen the beamwidths of mi-crostrip antennas, the antennas are normally mounted on curved surfaces [46].Taugchi et al. [47] and Yagyu et al. [48] analyzed the radiation characteristicsof a circular microstrip antenna inserted in the center of an oblate spheroidalconductor using the integral equation and the method of moments. Theyfound that the beamwidth in the .E-plane of the antenna became broaderthan with the normal circular patch antenna, and the beamwidth becomeswider as the antenna height is higher.

The effect of coating and radome on antennas is an interesting subject aswell, since antennas often have to be insulated from the surroundings, and thepresence of dielectric coating or radome can affect the radiation pattern. Forwire, cylindrical dipole, or spherical antennas, this issue has been addressedextensively in literature, while similar studies on spheroidal structures are lim-ited. The full-wave analyses of the spheroidal antennas in the uncoated case,dielectric-coated case, or the case where the spheroidal antenna is enclosed ina confocal radome, are presented in Chapter 6.

1.4 EM RADIATION IN DIELECTRIC SPHEROIDS

EM radiation in dielectric spheroidal structures was rarely discussed in pastyears, due to the difficulty of calculating the spheroidal wave functions ofcomplex arguments (for lossy dielectric objects). However, some applicationsare actually very important: for instance, the health effects of the human headexposed to EM energy emitted from mobile handset antennas. The humanhead can be approximated as a dielectric spheroid in order to calculate theEM power absorption distribution inside the head. Also, the distortions ofradiation patterns and the matched impedance of cellular phone antennasdue to the presence of the spheroid-shaped human head can be modeled usingspheroidal wave functions. Further, there are some analyses of radiated EM

8 INTRODUCTION

fields inside a prolate spheroidal human body model due to a loop antennathat is used as an EM therapy apparatus [49].

Perturbation theory [50] was applied to prolate spheroidal models to obtaininternal EM absorbed power distributions, but the convergence is generallyslow and the equations are valid only when the semiaxial lengths of the spher-oid are much longer than the wavelength [51-53]. Although perturbationtheory has the advantage of avoiding solution of the EM wave equations, in-stead requiring only the solution of equations that are similar to the staticequations, the method is valid only for low frequency or low ka (where k isthe wave propagation constant and a is the major semiaxial length of thespheroid).

The point-matching method was used by Ruppin to calculate the EM powerabsorption in tissue prolate spheroids irradiated by a plane wave [54]. The cal-culation was extended from the low-frequency region right into the resonanceregion, but it was still restricted to prolate spheroids of small eccentricities.

Iskander and Lakhtakia et al. studied the exposure of a prolate spheroidalmodel to the near field of a short dipole or a small loop antenna, using theextended boundary condition method (EBCM) [16,17,55,56]. The methodof solution involves an integral equation formulation of the transverse dyadicGreen's function, in which the irradiated field was expanded in terms of thespherical vector harmonics. However, formulas used in the EBCM have failedto provide fast convergent and accurate results for sources located at verysmall distances from the model. It is because the matrix size at a givenfrequency continues to increase with the decrease in the separation distanceand the results are less accurate as the distance of the observation point fromthe center of the inscribed sphere increased.

The investigation of the radiated fields inside a prolate spheroidal humanbody model due to a loop antenna was limited to the scalar analysis and the^-component of fields only [49]. Recent analysis of the prolate human headmodel using spheroidal vector wave functions was presented by Kang et al.[57]. In Chapter 7, some examples are provided as applications of spheroidalwave functions to the analysis of EM energy deposition and specific absorptionrate in a spheroidal human head.

1.5 OBLATE SPHEROIDAL MODELS

Although EM scattering by homogeneous oblate spheroid itself is an inter-esting subject, exploration of applications is still necessary: for example, thecalculation of rainfall attenuation of microwave signals due to oblate-shapedraindrops. The experimental photos and theoretical calculations show thatraindrops become oblate spheroidal in shape when the sizes of raindrops arelarge. In 1974, Morrision and Cross [58] computed the total cross section ofan oblate raindrop using the least squares fitting technique. They appliedthe perturbation method to a sphere equal in volume to the oblate spheroidal

SPHEROIDAL CAVITY SYSTEM 9

raindrop and of suitable eccentricity. Alternatively, Holt et al. used an inte-gral equation technique in their approach [59]. Aydin and Lure [60] used theextended boundary condition method to see the difference between millimeter-wave scattering and propagation in rain for oblate spheroidal raindrops andthat for spherical raindrops.

Li et al. implemented a simplified model using different expressions todescribe the upper and lower portions of a realistically distorted, nonaxisym-metric raindrop [61,62]. Based on this new model, they derived a total cross-section formula, which contains terms representing the zeroth-order approxi-mation (Mie scattering) and the first-order approximation (sphere distortionor perturbation theory) plus two additional analytical terms to account forspheroid-based distortion of the raindrops. The Doppler spectrum at verticalincidence was substantially affected by these different raindrop models. Seowet al. calculated the total cross scattering of oblate spheroidal raindrops usingthe T-matrix method [63]. Details of the analysis of rainfall attenuation ofmicrowave signals using oblate spheroidal raindrops are presented in Chapter8.

Attention has been drawn onto prolate spheroidal antennas, probably be-cause the prolate spheroid can be used to model more common antenna shapes.In the research by Taguchi et al. [47], they chose to work on the oblate case.In their work, the method of moments was used to obtain the radiation char-acteristics. However, the Mathematica package developed for the calculationof the prolate spheroidal antenna can easily be modified and applied to thoseoblate cases by using the simple replacement introduced in Chapters 2 and 3.

1.6 SPHEROIDAL CAVITY SYSTEM

The calculation of eigenfunctions and eigenfrequencies in spheroidal shapedcavities is usually made for acoustic problems. The EM resonant behav-ior of confocal spheroidal shells, spheroidal cavities, and a confocal prolatespheroidal cavity system has been considered earlier [1]. Recently, the EMeigenfrequencies in a perfectly conducting spheroidal cavity were determinedanalytically by Kokkorakis and Roumeliotis using the spherical eigenfunctionexpansion technique [64]. Although there was no need of using any spheroidaleigenvectors in their solution, the shape perturbation method and the result-ing relation they used were valid only when the spheroidal shape approachesa sphere. Although the spheroidal eigenvectors were used by Kokkorakis andRoumeliotis to calculate the eigenfrequencies in a spheroidal cavity and themethod was available for both TE and TM modes [65], the numerical resultswere applicable to those spheroids with small interfocal distances and theaccuracy deteriorated quickly for higher-order modes.

Kokkorakis and Roumeliotis also calculated the eigenfrequencies in concen-tric spheroidal-spherical cavities for both Dirichlet and Neumann boundaryconditions [66,67]. One of the techniques they used is shape perturbation,

10 INTRODUCTION

which is valid only when the spheroidal shape is close to a sphere [66]. An-other method is to express EM fields in terms of both spherical and spheroidalwave functions, associated with one another by expansion formulas [67]. Al-though the latter method can be applied to spheroids with large interfocaldistances, only the numerical results about small interfocal distances wereprovided in their report. The convergence of the expression becomes veryslow when interfocal distance grows, because convergence of the exponentialexpansion of eigenfrequencies depends directly on the interfocal distance andits power.

In Chapter 9, an alternative approach is proposed and is shown later tobe efficient. Detailed investigation of spheroidal cavity systems using theappropriate spheroidal vector wave functions is provided as well. Convergenceof the expression is faster and results on large interfocal distances are obtained.

1.7 SPHEROIDAL HARMONICS AND MATHEMATICA SOFTWARE

Computing facility and software pacakge availability have certainly improvedover the last four decades. Computer programs available to the public forcomputing the spheroidal harmonics and their eigenvalues are limited to thefollowing four: (1) a mathematical functions handbook published in 1992by Baker, who provided many useful routines/codes in C language for thecomputation of special functions including the spheroidal harmonics and theireigenvalues [68]; (2) a popular handbook series of routines/codes in Basic, C,Fortran, and Pascal published by Press et al. [3]; (3) two newly publishedhandbooks, one by Zhang and Jin [2] and another by Thompson [6], in whichthe authors published a large number of Fortran programs that are capableof calculating a wide variety of special mathematical functions to a reasonabledegree of accuracy; and (4) a published paper about spheroidal eigenfunctionsand harmonics with an attached Mathematica software package by Li et al.[19].

Although there have been many published papers on computations of theeigenvalues of spheroidal harmonics over the past several decades, most ofthem are applicable only for the real argument c. Also, so far, fewer com-mercial software packages, such as Maple, MathCad, and MatLab, areavailable for computation of these spheroidal harmonics. Recently, Li et al.presented a software routine package of the calculation of prolate (or oblate)spheroidal harmonics and eigenvalues on Mathematica software [19]. The al-gorithm of the package presented has also been verified for large and complexc values.

Mathematica is the world's fully integrated environment for technical com-puting. First released in 1988, it has had a profound effect on the way thatcomputers are used in many fields and by now has millions of users. SinceMathematica software contains a lot of symbolic and numerical built-in rou-tines with simple commands or kernels and the software package of the calcula-